Swampland conjectures and $f(R)$ gravity

First, the idea of swampland concerns to the class of effective field theories that cannot be completed to a string theory compactification.

With this in mind you could recognize that it is very difficult to realize $F(R)$ gravities as string compactifications. String theory indeed modify the Einstein-Hilbert action with higher Ricci scalar polynomial corrections and Lanczos-Lovelock terms; however, all those terms, with the exception of the Hilbert-Einstein leading term are suppressed by powers of the string length (see this blog post for more information). In other words, $F(R)$ gravities (possibly coupled to other force and matter fields) seem to be incompatible with perturbative string theory.

Of course the latter argument may be circumvented by adding some configurations of branes or other nonperturbative objects over some background, but it seems very unlikely because at any rate, you should be able to describe such background with some weakly variables and the problem rises again. Perturbative string theory seems to predict supersymmetric extensions of minimal Einstein's gravity minimally coupled to the dilaton (and other compactification space moduli) as its tree level theories.

Summary: $F(R)$ gravity theories are probably not compatible with string theory (=belong to the swampland).